Sum of Evenly Spaced Numbers via the Middle
A run of consecutive or evenly spaced numbers adds up to the middle value times the count (or pairs with a constant sum). Children first add short runs, then generalize to long sequences and grid relations where pairing collapses a tedious sum into one product. The insight reappears as the arithmetic-series shortcut.
grade 2–4 Base-ten numbersOperations Look for a PatternIdentify Subproblems
Progression (5)
g4 1. Sum a repeating number sequence Operations
g3 2. Sum consecutive numbers via the middle OperationsBase-ten numbers
g3 3. Consecutive integers around a middle value Operations
g3 4. Sum of evenly spaced numbers equals middle times count Operations
g4 5. Pair consecutive numbers with constant sum Operations
Bridges to competition (7)
Reasoning problems that extend this idea toward competition:
Simplify long sums using regularity Pattern & rule
Express a number as consecutive sums Number arrangement
Sum shaded numbers in an array Pattern & rule
Sum of a rectangle in a number grid Pattern & rule
Sum of consecutive numbers by pairing Pattern & rule
Simplify long add-subtract chains Invariant
Fast sums using pattern and symmetry Invariant